Free Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter Page A

other program to go through the boring ritual of checkmating. Although it lost every game it played, it did it in style. A lot of local chess experts were impressed. Thus, if you define "the system" as "making moves in a chess game", it is clear that this program had a sophisticated, preprogrammed ability to exit from the system. On the other hand, if you think of "the system" as being
    "whatever the computer had been programmed to do", then there is no doubt that the computer had no ability whatsoever to exit from that system.
    It is very important when studying formal systems to distinguish working within the system from making statements or observations about the system. I assume that you began the MU -puzzle, as do most people, by working within the system; and that you then gradually started getting anxious, and this anxiety finally built up to the point where without any need for further consideration, you exited from the system, trying to take stock of what you had produced, and wondering why it was that you had not succeeded in producing MU . Perhaps you found a reason why you could not produce MU ; that is thinking about the system. Perhaps you produced MIU somewhere along the way; that is working within the system. Now I do not want to make it sound as if the two modes are entirely incompatible; I am sure that every human being is capable to some extent of working inside a system and simultaneously thinking about what he is doing. Actually, in human affairs, it is often next to impossible to break things neatly up into "inside the system" and "outside the system"; life is composed of so many interlocking and interwoven and often inconsistent "systems" that it may seem simplistic to think of things in those terms. But it is often important to formulate simple ideas very clearly so that one can use them as models in thinking about more complex ideas. And that is why I am showing you formal systems; and it is about time we went back to discussing the MIU -
    system.

    M-Mode, I-Mode, U-Mode

    The MU-puzzle was stated in such a way that it encouraged some amount of exploration within the MIU -system-deriving theorems. But it was also stated in a way so as not to imply that staying inside the system would necessarily yield fruit. Therefore it encouraged some oscillation between the two modes of work. One way to separate these two modes would be to have two sheets of paper; on one sheet, you work "in your capacity as a machine", thus filling it with nothing but M's , I 's, and U 's; on the second sheet, you work "in your capacity as a thinking being", and are allowed to do whatever your intelligence suggests-which might involve using English, sketching ideas, working backwards, using shorthand (such as the letter `x'), compressing several steps into one, modifying the rules of the system to see what that gives, or whatever else you might dream up. One thing you might do is notice that the numbers 3 and 2 play an important role, since I's are gotten rid of in three's, and U 's in two's-and doubling of length (except for the M ) is allowed by rule II. So the second sheet might also have some figuring on it. We will occasionally refer back to these two modes of dealing with a formal system, and we will call them the Mechanic mode (M-mode ) and the Intelligent mode (I-mode ). To round out our mode with one for each letter of the MIU -system, I will also mention a fin mode-the Un-mode (U-mode ), which is the Zen way of approaching thing. More about this in a few Chapters.

    Decision Procedures

    An observation about this puzzle is that it involves rules of two opposite tendencies-the lengthening rules and the shortening rules . Two rules (I and II) allow you to increase the size of strings (but only in very rigid, pr scribed ways, of course); and two others allow you to shrink strings somewhat (again in very rigid ways). There seems to be an endless variety to the order in which these different types of rules might be applied,